Question

Integrate the function: \(x^2e^x\)

Answer 1

Let I = ∫x²e^x dx

Taking \(x^2\) as first function and ex as second function and integrating by parts, we obtain

I= x²∫e^x dx - ∫{\((\frac {d}{dx}x^2)\)∫e^xdx} dx

I = x²e^x - ∫2x.e^xdx

I = x²e^x-2∫x.e^x dx

Again integrating by parts,we obtain

I =x²e^x - 2[x.∫e^xdx - ∫{\((\frac {d}{dx}x)\).∫e^xdx} dx]

I = x²e^x - 2[xe^x-∫e^xdx]

I = x²e^x - 2[xe^x - e^x]

I = x²e^x - 2xe^x + 2e^x + C

I = e^x(x² - 2x + 2) + C